Generating Self-Affine Fractals by Collage Grammars

Self-affinity and self-similarity are fundamental concepts in fractal geometry. In this paper, they are related to collage grammars syntactic devices based on hyperedge replacement that generate sets of collages. Essentially, a collage is a picture consisting of geometric parts like line segments, circles, polygons, polyhedra, etc. The overlay of all collages in a collage language yields a fractal pattern. We show that collage grammars of a special type so-called increasing generalized Sierpinski grammars yield self-affine fractals. If one replaces the overlay by an intersection of all generated collages, the same result holds for decreasing generalized Sierpinski grammars. Here, the converse also holds: Every self-affine fractal can be generated by a decreasing generalized Sierpinski grammar, which provides a characterization of this class of fractals.